Hitting k primes by dice rolls

Abstract

Let $S=(d_1,d_2,d_3,\dots )$ be an infinite sequence of rolls of independent fair dice. For an integer $k\ge 1$, let $L_k=L_k(S)$ be the smallest $i$ so that there are $k$ integers $j\le i$ for which ${\sum }_{t=1}^j{d}_t$ is a prime. Therefore, $L_k$ is the random variable whose value is the number of dice rolls required until the accumulated sum equals a prime $k$ times. It is known that the expected value of $L_1$ is close to $2.43$. Here we show that for large $k$, the expected value of $L_k$ is $(1+o(1))k{\log }_{e}k$, where the $o(1)$-term tends to zero as $k$ tends to infinity. We also include some computational results about the distribution of $L_k$ for $k\le 100$.